Sets

Sets of various types form one of the core tools of verification for Dafny. Sets represent an orderless collection of elements, without repetition. Like sequences, sets are immutable value types. This allows them to be used easily in annotations, without involving the heap, as a set cannot be modified once it has been created. A set has the type:

   set<int>

for a set of integers, for example. In general, sets can be of almost any type, including objects. Concrete sets can be specified by using display notation:

method m()
{
   var s1 := {}; // the empty set
   var s2 := {1, 2, 3}; // set contains exactly 1, 2, and 3
   assert s2 == {1,1,2,3,3,3,3}; // same as before
   assert s1 != s2;  // sets with different elements are different
   var s3, s4 := {1,2}, {1,4};
}
   var s1 := {}; // the empty set
   var s2 := {1, 2, 3}; // set contains exactly 1, 2, and 3
   assert s2 == {1,1,2,3,3,3,3}; // same as before
   assert s1 != s2;  // sets with different elements are different
   var s3, s4 := {1,2}, {1,4};

The set formed by the display is the expected set, containing just the elements specified. Above we also see that equality is defined for sets. Two sets are equal if they have exactly the same elements. New sets can be created from existing ones using the common set operations:

method m()
{
   var s1 := {};
   var s2 := {1, 2, 3};
   var s3, s4 := {1,2}, {1,4};
   assert s2 + s4 == {1,2,3,4}; // set union
   assert s2 * s3 == {1,2} && s2 * s4 == {1}; // set intersection
   assert s2 - s3 == {3}; // set difference
}
   assert s2 + s4 == {1,2,3,4}; // set union
   assert s2 * s3 == {1,2} && s2 * s4 == {1}; // set intersection
   assert s2 - s3 == {3}; // set difference

Note that because sets can only contain at most one of each element, the union does not count repeated elements more than once. These operators will result in a finite set if both operands are finite, so they cannot generate an infinite set. Unlike the arithmetic operators, the set operators are always defined. In addition to set forming operators, there are comparison operators with their usual meanings:

method m()
{
   assert {1} <= {1, 2} && {1, 2} <= {1, 2}; // subset
   assert {} < {1, 2} && !({1} < {1}); // strict, or proper, subset
   assert !({1, 2} <= {1, 4}) && !({1, 4} <= {1, 4}); // no relation
   assert {1, 2} == {1, 2} && {1, 3} != {1, 2}; // equality and non-equality
}
   assert {1} <= {1, 2} && {1, 2} <= {1, 2}; // subset
   assert {} < {1, 2} && !({1} < {1}); // strict, or proper, subset
   assert !({1, 2} <= {1, 4}) && !({1, 4} <= {1, 4}); // no relation
   assert {1, 2} == {1, 2} && {1, 3} != {1, 2}; // equality and non-equality

Sets, like sequences, support the in and !in operators, to test element membership. For example:

method m()
{
   assert 5 in {1,3,4,5};
   assert 1 in {1,3,4,5};
   assert 2 !in {1,3,4,5};
   assert forall x :: x !in {};
}
   assert 5 in {1,3,4,5};
   assert 1 in {1,3,4,5};
   assert 2 !in {1,3,4,5};
   assert forall x :: x !in {};

Sets are used in several annotations, including reads and modifies clauses. In this case, they can be sets of a specific object type (like Nodes in a linked list), or they can be sets of the generic reference type object. Despite its name, this can point to any object or array. This is useful to bundle up all of the locations that a function or method might read or write when they can be different types.

When used in a decreases clause, sets are ordered by proper subset. To use sets in decreases clauses, the successive values must be “related” in some sense, which usually implies that they are recursively calculated, or similar.

You can test if a set is empty by comparing it to the empty set (s == {} is true if and only if s has no elements.)

A useful way to create sets is using a set comprehension. This defines a new set by including f(x) in the set for all x of type T that satisfy p(x):

   set x: T | p(x) :: f(x)

This defines a set in a manner reminiscent of a universal quantifier (forall). As with quanifiers, the type can usually be inferred. In contrast to quantifiers, the bar syntax (|) is required to seperate the predicate (p) from the bound variable (x). The type of the elements of the resulting set is the type of the return value of f(x). The values in the constructed set are the return values of f(x): x itself acts only as a bridge between the predicate p and the function f. It usually has the same type as the resulting set, but it does not need to. As an example:

method m()
{
   assert (set x | x in {0,1,2} :: x + 0) == {0,1,2};
}
   assert (set x | x in {0,1,2} :: x + 0) == {0,1,2};

If the function is the identity, then the expression can be written with a particularly nice form:

method m()
{
   assert (set x | x in {0,1,2,3,4,5} && x < 3) == {0,1,2};
}
   assert (set x | x in {0,1,2,3,4,5} && x < 3) == {0,1,2};

To reason about general, non-identity functions in set comprehensions, Dafny may need some help. For example, the following is true, but Dafny cannot prove it:

method m()
{
   // assert {0*1, 1*1, 2*1} == {0,1,2};  // include this assertion as a lemma to prove the next line
   assert (set x | x in {0,1,2} :: x * 1) == {0,1,2};
}
   assert (set x | x in {0,1,2} :: x * 1) == {0,1,2};

To help Dafny prove this assertion, you can precede it with the assertion assert {0*1, 1*1, 2*1} == {0,1,2};. This lets Dafny figure out both assertions.

Without care, a set comprehension could prescribe an infinite number of elements, but a set is only allowed to have a finite number of elements. For example, if you tried writing set x | x % 2 == 0 as the set of all even integers, then you would get an error. (If you really want an infinite set, use the iset type instead. For example, iset x | x % 2 == 0 is legal in ghost contexts.) To ensure that set comprehensions give rise to finite sets, Dafny employs some heuristics. When creating sets of integers, this can be done by bounding the integers in at least one conjunct of the predicate (something like 0 <= x < n). Requiring a bound variable to be in an existing set also works, as in x in {0,1,2} from above.